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Losing Weight by Gaining Edges

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Algorithms - ESA 2014 (ESA 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8737))

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Abstract

We present a new way to encode weighted sums into unweighted pairwise constraints, obtaining the following results.

  • Define the k-SUM problem to be: given n integers in [ − n 2k,n 2k] are there k which sum to zero? (It is well known that the same problem over arbitrary integers is equivalent to the above definition, by linear-time randomized reductions.) We prove that this definition of k-SUM remains W[1]− hard, and is in fact W[1]-complete: k-SUM can be reduced to f(k)·n o(1) instances of k-Clique.

  • The maximum node-weighted k-Clique and node-weighted k-dominating set problems can be reduced to n o(1) instances of the unweighted k-Clique and k-dominating set problems, respectively. This implies a strong equivalence between the time complexities of the node weighted problems and the unweighted problems: any polynomial improvement on one would imply an improvement for the other.

  • A triangle of weight 0 in a node weighted graph with m edges can be deterministically found in m 1.41 time.

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Author information

Authors and Affiliations

  1. Computer Science Department, Stanford University, Stanford, CA, USA

    Amir Abboud, Kevin Lewi & Ryan Williams

Authors
  1. Amir Abboud
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  2. Kevin Lewi
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  3. Ryan Williams
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Editor information

Editors and Affiliations

  1. Massachusetts Institute of Technology, Cambridge, MA, USA

    Andreas S. Schulz

  2. Karlsruhe Institute of Technology (KIT), Kaiserstruhe 12, 76131, Karlsruhe, Germany

    Dorothea Wagner

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Abboud, A., Lewi, K., Williams, R. (2014). Losing Weight by Gaining Edges. In: Schulz, A.S., Wagner, D. (eds) Algorithms - ESA 2014. ESA 2014. Lecture Notes in Computer Science, vol 8737. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44777-2_1

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  • DOI: https://doi.org/10.1007/978-3-662-44777-2_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-44776-5

  • Online ISBN: 978-3-662-44777-2

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